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Call to revisit the Mesoamerican Calendars
The count that is called the Real Calendar
André CAUTY
CELIA
… en la cuenta que llaman calendario verdadero cuentan 365 días,
y cada cuatro años contaban 366 días.
Sahagún
Only a detail separates Mayanists and Nahualists. For the first, the
Maya calendar contains exactly 18 9801 kin ‘days’. For the latter, the Aztec
calendar consists of 52 xihuitl years whose total in days – determined by the
type of the year – is supposed to be the same: 52 x 365. A supposition to be
confirmed or invalidated. Showing that the CR of 18 980 days and the Aztec
xiuhtlalpilli of 52 years are two different types of calendars,2 this synthesis
invites the reader to reconsider the traditional theory according to which the
sharing of a mutual calendar is a defining trait in the concept of Mesoamerica
(Kirchhoff 1943).
All Mesoamerican calendars seem to be defined by a common
structure of a product determined by cycles whose root is the combination of
an almanac of divination of 13 x 20 days and a vague year of (18 x 20) + p
days.3 The calendar expressions in fact combined in the Mesoamerican hic et
1 GCM (260, 365) is one fifth of the product tzolkin x ha’ab = 94 900. It is also 949 uinal ‘months’, 73
tzolkin or 52 ha’ab. Its double, 2 CR, equals 65 cycles of Venus.
2 In spite of a probable common origin and important similarities such as the order of Year Bearer
sequencing and that of eponym changes.
3 Mayan tzolkin or Aztec tonalpohualli of 260 days; Mayan ha’ab and Aztec xihuitl of 365 or 360 + p
days, with the question : p is it always and everywhere equal or not to 5 ?
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nunc are, however, visibly different between the two peoples. Each group of
cities followed its own rules for dating events and distinguishing days.
Common to all Mesoamericans, the ‘almanac’ dates are confirmed4
from the middle of the VIIth century BC until the colonial period. These are
expressions of the form αX, where α follows a cycle of 13 successive
integers, and X a cycle immutably ordered of 20 day-glyphs. Or, a cycle
which is the product of 260 dates, established with the order: s(αX) = [s1(α),
s2(X)] where the 's' are the 'successor' functions of the cycles being
considered. Nearly absent with the Aztecs, the vague year dates are later and
are only clearly attested to with the Mayas.5 These are of the form βY,
wherein β follows the cycle (0, 19) and Y the ordered cycle of 18 named6
Months, veintena, of 20 days, and of the named complement Uayeb. Thus a
set of 365 days/dates for the 365-day year, ha'ab, equipped with the order s’:
s’(Y) = [s3(, Y] for each Y ≠ Uayeb et β < 19
s’(19Y) = [s3(19), s4(Y)] = [0, s4(Y)]for each Y ≠ Uayeb and β = 19
s’(β Uayeb) = [s3(Uayeb] for each  < 4
s’(4 Uayeb) = [s3(4), s4(Uayeb)] = 0 Pop
1. The product tzolkin x ha'ab of the Maya and the cycle of Bearers
The CR7 Maya is the ordered product tzolkin x ha'ab whose elements
are couples (αX, βY). The study of the mathematical properties of the ordered
products of ordered cycles has shown (Cauty 2009: 20-30) that the conjunction
of three factors,8 along with the fact that 260 and 365 are multiples of the same
4 The older are anthroponyms (Urcid, Pohl and others).
5 The 0 Yaxkin of the Leyden plaque (15/09/320) is among the first evidences of Y dates. The latter, from
the Post Classic, are in the codex (Dresden and Paris).
6 In colonial Yucatec, for example: Y covers the following (Pop, Uo, Zip, Zodz, Tzec, Xul, Yaxkin, Mol,
Ch’en, Yax, Zac, Ceh, Mac, Kankin, Muan, Pax, Kayab, Cumku) that closes the complement Uayeb.
To know the date Y of a day is equivalent to knowing its position  in the ha’ab: the cycle of the ha’ab
dates ordered by s’ is isomorphic to a group of 365 natural integers fitted with the natural order of
integers.
7 Its Mayan name is unknown. But its value of 2.12;13.0. is established. Mayanists use the expressions:
Calendar Round, Wheel Calendar, Ritual Calendar.
8 Classic Mayan CR. Let S be that difference which distinguishes, like a signature, the CR from its clones.
F1: the rules for formulating the expression composed of dates (tzolkin, ha’ab and CR), F2: the type of
numeration of the numbers in the pairs XY, and F3: the relative position (defined for instance by the
origin date 4 Ahau 8 Cumku) of tzolkin and ha’ab at the moment of starting the CR. To distinguish the
XY expressions that belong to the Classic Mayan CR, and not confuse them with an X*Y expression
which does not belong to the CR, we can calculate the difference S(XY) = │x - │ (modulo 5) and
compare with the difference S(4 Ahau, 8 Cumku). In case of a tie, the date (XY) belongs to the Classic
CR, otherwise it is a starry date that does not belong to the Classic Mayan CR. Let S be that difference
which distinguishes, like a signature, the CR from its clones
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number,9 has two important consequences. First, to limit the number of CR
dates to 18 980. And to produce, in addition, cyclic events10 that are resumed
by the theorem: Whatever the integer P, the almanac date of Pth day of the
vague year is of the form αXP, where α follows the cycle of 13 integer almanac
dates, and where XP follows a class, modulo 5, of four X day names. Each day
of the 365-day year is thus associated with 13 x 4 = 52 almanac dates who
characterize it and succeed one another year after year according to the law:
s(αXP) = s(α)s(XP) = [(α +1), (XP +1)] = [(α +1), (XP +5)]. The value P = 0
defines the 1st day of the 1st month of the Mayan year, the New Year. Applied
to this day, the theorem states, first, that the Mayan New Year is associated
with four11 tzolkin XP day names. In other words, the second consequence says
that each New Year date αXP distinguishes and defines one and only one ha'ab
year in the group of 52 years that make up the CR. The system of dates αXP
supplied a practical means12 to distinguish, define and name the years of a CR:
by making αXP the eponym for the year.
2. Tonalpohualli, xihuitl and Aztec eponyms
Like all Mesoamericans, the people submitting to the Aztec Triple
Alliance used the αX dates of the almanac. Public or private, their lives were
also organized in cycles of 52 xihuitl years, each consisting of 18 veintena
months of 20 days and a period, Nemontemi.13 However, the Aztecs have not
left written βY dates prior to the arrival of the Spanish, with the help of the
9 Their GCD 5, greatest common divisor.
10 It is the occasion to celebrate, each year among the Maya, the change of Year Bearer; and every 52
years among the Aztecs, the Binding of the years (xiuhtlalpilli) and the New Fire.
11 The entities associated with these 4 names are called the 4 Year Bearers: Ik, Manik, Eb and Caban in
classic Mayan period.
12 Probably unused in the classic period because the dating methods were particularly redundant.
Especially in the use of the solemn public use of stelae and monuments recounting the glory of Mayan
cities and leaders in the enameled texts of dates given in the CR system, but also in the Long Count
ci(Pi), in Lunar Series, and other cycles as well. For example: CL 13-baktun 0-katun 0-tun ; 0-uinal
0-kin and the date CR 4 Ahau 8 Cumku from the Stela E of Quiriguá (Guatemala, 771 AD).
13 Except for those who attribute to Mesoamericans the use of a 366-day leap year, sources state that the
Nemontemi contained 5 unlucky days, unnamed, sleeping… In Historia general de las cosas de Nueva
España, Sahagún gives a list of 18 expressions traditionally accepted as that of the 18 month names
although they are “very different from the point of view of their syntactical structure: 'there are gifts of
flowers', 'the trees sit up straight', ‘ little watch’, ‘skinning people’…” (Launey 2009: personal communication). Sahagún also gives the associated divinities, the name of the period Nemontemi, as well as the
position (at the period of the months of the Julian calendar. The second veintena, Tlacaxipehualiztli,
went from March 4 to 23; dedicated to Xipe Totec, it was characterized by the skinning of people. The
list is attested to by multiple sources, modulo differences: the number of days and the position in the
year of the period Nemontemi, the month that begins the year and subsequently the numeration of the
months. The month Quecholli, for example, is generally the 14th month, but it is the 13th for Rámirez.
In two different texts (manuscript 215 and Historia antigua de Mexico) Ixtlilxochitl begins/ends the
year in: Atlcahualo/Nemontemi and Atemoztli/ Panquetzalztli (Roulet 1999).
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glyphs of the indigenous pictographic writing. Only certain texts from the
colonial period transcribed the detailed Aztec forms for a handful of event
dates that were critical for both Worlds into the Latin alphabet;14 for example
“8 Ehecatl 9 Quecholli of 1 Acatl” for Cortez' entry into Mexico City. We
thus have only a small body of βY dates of the type 9 Quecholli. To which
we may add more vague indications stating, for example, that the twenty
monthly ceremonies took place at the beginning, the middle or at the end of
the veintena. Usually not recorded, the rank within the month remains
uncertain when we have it available. The reference 9 Quecholli for example,
or the indications first / middle / last day of the month Y do not allow
confirmation that the days corresponding to these dates were, indeed, at the
positions 9, 1, 10 and 20 of the month Y. Why is this? Because the sources do
not say if the Mesoamericans counted their days in the same manner,
especially if they began with the same number. However, we are certain that
the ways of counting were diverse: the Mayas wrote the numbers from 0 to
19, the Spanish numbering the days from 1 to 31 and the Tlapanecs from 2 to
14. From which are derived two deductions:
1) The CR of 18 980 days distinguished and dated by as many expressions
αXY is not a tangible reality in the space/time of the Triple Alliance.
2) We can however, from a 9 Quecholli for example, undertake the
reconstruction of the dates Y of a xihuitl.15 Besides the formula of xihuitl
and its invisible dates,16 the Aztecs inherited, on one hand, knowledge of the
duration (52, in number of years) of the CR and, on the other hand, of the
effects of the conventions that structured it and which began the Bearer cycle.
This heritage takes into account the representations of successions of xihuitl
and of xiuhtlalpilli17 and most importantly the habit of distinguishing and
noting the years by the ordered succession of their eponyms αXP18 like that of
the folio 2r of Mendoza (annexe) going from the year 2 Calli to the year 13
Acatl in passing by 2 Acatl signaled as the year of the celebration of the New
fire. Thus, the Aztec date for the day of an event is an expression (X,XP),
14 The most sure witness reports concern Cortez’ entrance into Mexico City (08/11/1519), the Night of
Sorrows when he is driven out, and the destruction of the Temple of Mexico (Tena 1987: ch. IV).
15 In a more or less credible manner according to the suppositions retained, beginning with that of the
number of days attributed to the xihuitl (365 or 366?). Or to reconstruct the Y dates of the 52 years of a
xiuhtlalpilli, but this with yet more uncertainty.
16 The Y dates of type 9 Quecholli that we never see written in indigenous glyphs.
17 For example, that of the 266 years of the Azoyú Codex and the xiuhtlalpilli of the Durán Codex.
18 XP are the 4 days, linked to the Bearers among the Mayas. Among the Aztecs, these are the days: Calli,
Tochtli, Acatl and Tecpatl whose Mayan equivalents are Akbal, Lamat, Ben and Edznab, witnessed
in the Dresden Codex, but which are neither Ik, Manik, Eb and Caban presented in Classical Mayan
nor Kan, Muluc, Hix and Cauac recorded in the Colonial Period by Landa and the Madrid Codex.
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where X is the tonalpohualli date of the day of the event and XP that of the
eponym day of the year in which it occurs. This mode of dating does not
result in 18 980 dates as in the case of the Mayan CR, but only 260 x 52
possible different expressions.19 By its construction, it is ambiguous: 260
dates X are not sufficient to distinguish the 365 days20 of a xihuitl; and 13
520 pairs (X,XP) are not sufficient to date the days of an Aztec Century of
18 993 days – in verdadero (Sahagún) and Julian calendars – or of 18 980
days if one decides to identify it21 with the Mayan CR.
3. Difficulties and differences
Identifying Mayan and Aztec calendars is a habit22 whose principal
fault is to hide a specific difference: only the Classical Mayas commonly
wrote Y dates of the annual ha’ab calendar. When it is noticed,23 the
difference – the inscription of the eponym vs. the ha’ab date – it is
sometimes denied by an interpretation that reduces it to a question of
abbreviation or preference.24 Small cause, big consequence: this stylistic
nuance would give, to the Nahuas, 13 520 dates (X, XP); and to the
Mayas, 18 980 dates (X, Y). No, Aztec formulas such as “8 ehécatl of 1
ácatl” and Mayan such as “4 ahau 8 cumku” are not abbreviations denoting
the same calendar reality.
These are two types of dates to be analyzed //8 Ehecatl/1 Acatl//
and //4 Ahau/8 Cumku//, of very different components. The 1rst
19 Not a proper part of the product, though 52 and 260 are divisible by 4 (see the CR), because the law of
20
21
22
23
24
succession (‘linear enumeration’, type: 1 January, 2 January, etc.) defined in tonalpohualli x
xiuhpohualli – s(X, XP) = [s(X), XP] as long as the xihuitl is not passed, otherwise s(X, XP) =
[s(X), XP+1] – is not the same as that defined in tzolkin x ha’ab (‘diagonal enumeration’, type:
Monday 1, Tuesday 2, etc.).
It is always possible to render the calendar unambiguous by taking an additional cycle, for example, that
of the 9 Lords of the Night.
Decision leading to the creation of 52 x 105 double dates X to add, in an order to be specified, to the
260 already used in each xihuitl year of the Aztec century.
“El calendario mesoamericano era el resultado de la combinación entre un ciclo de 365 días, llamado en
náhuatl xiuhpohualli o “cuenta del año” (ha’ab en maya), y otro ciclo de 260 días, llamado en náhuatl
tonalpohualli o “cuenta de los días” (tzolkin in Maya) […] Se requería el transcurso de 18 980 días
nominales, equivalentes a un “siglo” de 52 años, para que se agotaran todas las posiciones posibles de
un día cualquiera del tonalpohualli dentro del xiuhpohualli, y viceversa” (Tena 2000: 5).
Which it is not always, because the habit of transcribing everything (tzolkin date, ha’ab date, period,
etc.) in the same manner neutralizes and renders invisible many of the differences noted by the scribes.
“Tanto los nahuas como los mayas utilizaban una fórmula abreviada para los fechamientos, pues
ordinariamente no se mencionaban en forma completa todos los elementos que intervenían en una
fecha, a saber: el día del tonalpohualli, el ordinal del día dentro de la veintena, y el año. Los nahuas
preferían enunciar sólo el día del tonalpohualli y el año; decían, por ejemplo, 8 ehécatl de 1 ácatl. Los
mayas, en cambio, sólo enunciaban el día del tzolkín y el ordinal de la veintena; decían por ejemplo: 4
ahau 8 cumku” (Tena 2000: 5. AC underlined in bold).
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component does not introduce any difference: for both an Aztec and a
Maya, it is an X almanac date. But the second components are, both in
nature and form, different. In Nahuatl: another tonalpohualli date XP, but
in Mayan: a ha’ab date βY. There is no isomorphism.25 But rather a Type B
elder who led the Classical Mayas to write dates (X, Y) and to imply the
eponym XP and a Type A who, during the Postclassical, led people to
write dates (X, XP) and to imply the date [Y]. The formulas are:
Type A = (X, Y), [XP] among the Mayas26
Type B =X, XP), [Y] among the Aztecs.27
4. Reflections and conclusions
The non-isomorphism of the Aztec and Mayan calendars possibly
has its root in the fact that only the Mayas used the long Count Σci(Pi) and
the CR datesXY) in a joint and immutable manner. This usage led, in
effect, to the rigorous synchronous maintenance of the tzolkin and ha’ab
cycles and to never change the number of days of the 365-day year. The
real benefit of this rigor was, of course, the possibility of making precise
calendar calculations28 with the help of the Multiplication Tables29 and the
Date Tables that we find so abundantly in the Codex and which served to
accomplish the challenges of modular Mayan arithmetic.30
Without this rigor and these tools, the scribes would undoubtedly not
have been able to simulate the return of eclipses, correct the shortcoming of
the Venusian leap year or record the embellished narrative tales of a network
25 In consequence of this non-isomorphism: the contrast between the Mayan facility to find the eponym
26
27
28
29
30
XP from the date X Y of an event, and the difficulty for a Nahualist to find the date Y from the
date (X, XP). A Mayan dateX Y defines one and only one day of the CR. The position in the
ha’ab of the day Y is “→Y” in spoken protractive numeration. Subsequently, the eponym (tzolkin
date of the day 0 Pop) sought is given by XP = X – (→ Y). For example: 7 Eb is an eponym for
the year of 4 Ahau 8 Cumku. Another facilitating factor for the Mayanists is the richness of redundant
elements in the calendar expressions.
Whose calendar system imposed the constraint of co-occurrence on the marked components (tzolkin
date and ha’ab date) and implying the redundant eponym.
Whose calendar system imposed writing the eponym XP of the year, but no constraint of cooccurrence on the tonalpohualli dates (marked) or the xihuitl dates (never written).
In practice, to one day. The fact that the scribes used all sorts of cycles may be interpreted by saying
that they calculated in rings Z/nZ or in classes of appropriate wholes modulo n. Clocks give us a
familiar image of this type of calculation which lowers by n any number that reached or surpassed this
value, because the hour hand makes additions modulo 12: if it starts from 7 o'clock, for example, and
ten hours must be added, it will not mark 17 00 hours, but 5 o'clock.
Containing at times, in the position of intruder, non-multiple numbers serving to correct the calendar
deviation in vague years, like that of Venus of the Dresden Codex.
Given 2 dates x, y (of the Mayan CR to clarity things) and one translation t (in whole number of days),
solves the 3 equations t(x) = y according to whether the unknown is x, y or t.
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of dates and numbers of distance that celebrated the grandeur of gods, cities,
kings... Otherwise stated, it is indeed the rule ROCm (Cauty 2009b:10-12),
consequence of the circumstances described above, and cause of the
specificities expressed in § 1 by way of the theorem which serves as the basis
of the possibility of calculating to the day, and which puts it into practice with
the constraints of co-ocurrence imposed on couples of the product tzolkin x
ha’ab. It will be further shown that the redundancy brought by the use of the
CL31 acted as a code detecting, even correcting, errors.
For the Aztecs, the product (X, XP) of the dates tonalpohualli x
xiuhpohualli is not limited by the factors F1, F2 and F3 because the Y dates
were not or no longer written. Also, in the absence of the Long Count,32
eventual discrepancies or other calculation errors become difficult not to
detect. The Century, xiuhtlalpilli, was thus freed from the functional
obligations which the ROCm Rule imposed upon the product tzolkin x
ha’ab. Due to this, the cycles X and XP were like 'free spining gears' in
relation to one another. The Almanac had sorted out its 260 X dates, but
it needed something or someone to increment and count the eponyms XP
and to maintain the 52-year cycle that no longer rigidly tied to the xihuitl.
Colonial or not, the sources do not explain how
Mesoamericans without Y dates went about knowing
when to increment the eponym. In principle an Aztec
was not obliged to change xihuitl like a Maya would
change ha’ab, that is to say: on the passage from the
365th and last day of the year n-1 dated 4 Uayeb to the
1rst day of the year n dated CHUM/0 Pop.33 Numerous
figures revealing the persistence of the number and
of the order of succession of the 52 years of a xiuhtlalpilli prove that the
tradition of the 365-day year was maintained (Duran Codex).
However, the insistent indication – stating, without proof, that the
Natives corrected the vague year, that they possessed a 'real' calendar to
31 But also the Distance Numbers, the lunar Series, and other cycles, like that of the Lords of the Night (or
patrons of the Underworlds).
32 The 'small' generative capacity of the Aztec written numeration system and its repetitive form are
elements unfavorable to the writing of large numbers and the composition of tables of multiples, that
partially explains this absence of Aztec Long Counts.
33 Which amounts to changing the manner of counting the days, following the end of the installation of the
New Year. Between 0 and 1 Pop or between 1 and 2 Pop or 2 and 3 Pop according to the strategies of
enumeration of the ranks and the counting of days that we know to have started at 0, 1 or 2 according
to the peoples and the periods.
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which they added a 366th day every four years – proves two things relative
to the Postclassic and Colonial calendar customs. Firstly, that they knew
the 365-day year. Secondly, that it seemed to remain in phase with a
tropical year of 365.25 days or with a year defined by a whole number of
days but of variable length.34
The Aztecs could, of course, recognize the changing of seasons or
of years. A reasonable hypothesis is thus to suppose that the changing of a
year and of the eponym XP could be decided following the appearance of
a designated natural or astronomic sign: the passage of the Sun at the
Zenith of a site,35 a Solstice, a passage at the meridian of Miec/Tianquitzli
(Pléiades), the Bridge of Turtles...
In Los observatorios subterráneos,36 Rubén B. Morante López
evaluates the research and the measures taken since the eighties by Aveni
and Hartung (1981), Anderson (1981), Broda (1986, 1991) and Tichy
(1980, 1992) in the subterranean observatories, notably that of
Xochicalco.37
The authors record the days upon which rays on light enter into the
chamber of the observatory, and the days upon which they do not. The
results show that those who conceived the observatories constructed the
chimney in such a way as to divide the year in two: one part during which
the chamber received the Sun's rays, and a part in which it did not.
According to the measurements taken in 1988 – 1992, the first period
began on April 30 (once on May 1) and the second on August 13. These
dates divide the year into a portion of 105 days and one of 260 days (261 in
leap years). The 105 day portion is centered on the Solstice38 of June 21:
34 Consequently, the dates (eponym, nth day of the month, New Year, etc.) do not stray in the seasons like
the Mayan 0 Pop should – in spite of the affirmations by Landa who fixed the Mayan New Year on the
Christian July 16th, and asserts that the scribes added a 366th day to the year every four years. This type
of year more or less elastic is known: it is the festive year (18 x 20 + p) or the Catholic liturgical year
which has 52 or 53 weeks.
35 In the intertropical zone, the Sun passes twice at the Zenith (before and after the Summer Solstice),
and each passage is easily noted by the absence of shadow for objects held vertically (stelae, for
example). One may also follow the rising and setting of the Sun in relation to reference points of
the horizon or of the city, etc. All of this provides the means to elaborate an annual calendar in
phase with the tropical year.
36 On-line: http://www.uv.mx/dgbuv/bd/pyh/1995/2/html/pag/index.htm, the article appeared in 1995 in
La palabra y el hombre.
37 Are noted (Morante Lopez 2001:48) the subterranean observatories of Teotihuacan (200 AD), Monte
Alban (400 AD) or Xochicalco (700 AD).
38 Framed by the 2 passages at the Zenith whose date depends on the latitude of the site.
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April 30 + 105 + August 13, August 13 + 260 = April 30,39 April 30 + 52 =
June 21, June 22 + 52 = August 13.
The Aztecs thus disposed of a sort of clock or of calendar giving,
live and continuously, the progression of the days and seasons of the
tropical year:
01/V
Solstice
21/VI
12/VIII 13/VIII
30/IV
The interest of the experiment of Xochicalco is to reveal the
following points:
Certain Mesoamericans constructed heliographs (markers of rays of light)
that gave the beginning and end of two periods. – A lighted period that
includes the principal bearings of the tropical year (Summer Solstice and
passages at the Zenith) and whose length of 105 days enjoyed undeniable
numerological properties. For example: 105 = 5 x 20 + 5 = 2 x 4 x 13 + 1. A
period of shadow lasting 260 days equaled the length of the divinatory
almanac and catches up with, in four years, the delay of one day that the
vague year has on the tropical year. The period of shadow lasts 260 days in
a normal year and 261 days40 in leap years.
Vague Year No. 1
105 days
260 days
Vague Year No.2
105 days
260 days
Vague Year No. 3
105 days
260 days
Vague Year No. 4
105 days
261 days
Vague Year No. 5
105 days
260 days
Vague Year No. 6
105 days
260 days
Disposing of such a heliograph, the kings and priests have no need
of a calendar of the 365-day year, nor even to mark the 365 passing days.
Because, in order to know at which point was the tropical year, it was
sufficient to go to the observatory to read and interpret what the Sun's rays
revealed in this sacred place. And to decide, for example, to increment the
Book of Years or to celebrate the New Fire. Without possible error. But in
a manner quite distant from the calendar and computational habits of the
Classical Mayas.41
39 August 13 + 261 = April 30, during leap years.
40 Because d(August 13, April 30/May 1) = 260/261 according to whether February counts 28/29 days.
41 At best linked by a cryptomorphism the calendars (X, Y) and (X, XP) do not even speak the same
language. The 52-year Aztec Century is an adjustable simulation of the solar year, while the Mayan CR
is an untouchable arithmetic model made to distinguish and to define each of the 18 980 days of the
most typical Mesoamerican temporal cycle. At the price of losing the dates during the seasons and not
claiming a 'true calendar.'
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During the Postclassic and Colonial period, most of the peoples
contented themselves with the dates (X, XP) and to follow in parallel the
course of the months of year and the rhythm of the 20 monthly
celebrations. In the areas mixed by contact with the Spanish, the Natives
had every interest in hiding their attachment to the sacred almanac and to
the eponym cycle, largely stigmatized as Satanic works. One way to do so
consisted of becoming a user of the vague solar year calendar. Ha’ab and
xihuitl are, indeed, much close to the calendar of the Spaniards, even if
they contain 18 months of 20 days instead of 12 months of 30 days on
average. The Colonial sources contain Indian calendars that seem to stem
from this state of things. These are the annual calendars that respond like
the Maya ha’ab or the Aztec xihuitl to the formula (18 x 20) + 5. But, that
differ from it in the manner of writing the 20 dates of a Y month. At a first
analysis, these calendars reveal three types of different situations:
1) The dates of the Mayan months of the Classic Period are of the form Y
and, without going into the details of the representation of Uayeb, we may
represent the ha’ab calendar by a table of 18/19 columns labeled by the 18
Y names42 of the months plus Uayeb, and filled the 360/365 cells of the
array by the sequence of the twenty first natural integers which semiotize
the twenty numbers  of the days in the month.43
2) For the Mayas of the Postclassic or for the Colonial periods, the months
remained unchanged and continued to label the columns of the chart, but
the  positions are no longer written.44 The monthly columns are, however,
filled and learned by the sequence of the 20 X dates of the days of the
month, to which the rows  bring a strong character of trecena. More
precisely, the columns are filled by the dates iXj where the indexes i and j
vary from 1 to 20, modulo 13 (for the rows) and modulo 20 (for the X
names).45 In the example of the Calendario de los Indios de Guatemala,
the 20 lines of the days of a month are additionally numbered from 1 to 20
which are said to be the positions of the days in the month. This is
42 The Y names of the months evidently change with the languages and the periods, but all of the lists
contain, with the exception of one translation, the same months in the same order of the type Pop, Uo,
etc., Cumku, Uayeb.
43 These 20 rows vary from 20/0 (TI’HA’B/CHUM) to 19.
44 Sometimes doubled by the number y of its position in the succession of the months of the ha’ab.
Sometimes translated into Nahuatl or another language, and sometimes doubled by a description in
Spanish or Latin. For example, in the Calendario de los Indios de Guatemala, 1685, Cakchiquel,
http://famsi.org/research/mltdp/ item57 we have: Mes n° 10 Rucactoi
45 Reading horizontally, the Xj are constant, and the i in arithmetic progression with a common
difference of 7 (modulo 13), so that, in 1 out of 2 columns, they are in the natural order of integers.
CAUTY A. : Invitation to revisit the Mesoamerican Calendars…
125
evidently a notation made by/for a Spaniard, and not a notation which
refers to the or to the Y dates in the Mayan ha’ab of the Classic Period.
3) For the non-Mayas of the Post Classic or Colonial Periods, the months
are not identified by their Y name,46 but by a scene or a description which
seems to vary locally. A bit like the well-known expression Le temps des
cerises allows a French person to identify the month of June.47 As in 2), the
columns of months are filled with the dates (tonalpohualli) of the form iXj.
As shown by the examples of Annexes, it results from 1) and 2) that
the Mayan ha’ab remained isomorphic to a 365-day year, which amounts
to saying that the ROCm rule remained in use. No longer recording the Y
dates allowed, however, the tolerance of minor deviations: to change the
initial positions of the tzolkin and ha’ab cycles in relation to one another at
the starting moment of the CR. One such deviation reveals itself by a
change in the role of the Bearers or the necessity to change the numerical
value of the difference S from note 8. But it does not modify the
organization of the CR, which remains a calendar of 18 980 dates. In this
case, it would be legitimate to reconstruct, from a fragment, the 365 dates
of the annual calendar, or the 18 980 of a CR.
In the cases 2) and 3), taking into account the bias introduced by the
Colonizer/Colonized contacts, and the imprecision already signaled
concerning the eventual length of the xihuitl (365? 366? 365.25?), the dating
(X, XP) does not, on its own, allow the reconstruction, from a 52-year
Aztec Century, of a 18 980-day Calendar Round isomorphic to a Mayan CR
cycle.48 From whence the difficult question of an eventual corrective49 of the
delay of the ha’ab or xihuitl calendar over the tropical year.
Revisiting the Mesoamerican calendars would consist of crossing a
typology of the calendars50 with a typology of the situations and users.51
46
47
48
49
Month’s sign, or the y position of the month Y in the 18-month year.
For an example: http://www.lunacommons.org/luna/servlet/view/all/who/Tovar.
Marked by 52 XP dates in the pages 34-37 of Madrid codex (dates XP in p. 54-57 of the Dresden).
Because correcting the vague year amounts to increasing the duration of a period and redistributing the
normal/increased periods. This provokes a substantial delay likely to explain, for example, the diversity
of the roles of the Year Bearers or the dates (X, [Y) of the Chilam Balam de Tizimin which
contradicts the ROCm of the Classic.
50 On the basis of respect of the ROCm or of the 4 other solutions of equation │x –  │= S (modulo 5) of
the deviations of these rule, we can distinguish, for example: calendars of the type Aztec Century or
calendario verdadero of 18 993 days (52 xihuitl of 365/366 days), calendars of the type CR Mayan
Classic of 18 980 days or its four clones, characterized by the roles P0, P1, P2, P3 and P4 of the Year. A
deviation from the ROCm rule may be provoked by: a change in the variation interval of the rows , a
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AMERINDIA n°36, 2012
This work could cast new light upon the diversity of the roles of the
Bearers, eponyms and calendars, and show that the kings of cities with
heliography could have, without writing Y dates, maintain in phase the
succession of the xihuitl and that of the tropical years. Otherwise stated,
here or there, the Calendario verdadero may well have had a pre-Hispanic
reality. Certainly, different both from the reality of Julian and Gregorian
calendars and that of reinterpreted Mesoamerican calendars, in the mestizo
zones of interaction between the two Worlds, by and for the Colonial and
Evangelist institutions of the Europeans.
References
ALVA IXTLILXOCHITL, Fernando de
1977 Obras Históricas. Gorman E. O’ (ed.), Tomo 2. México:
UNAM/Instituto de Investigaciones Históricas.
AVENI, Anthony F.
1981 Tropical Astronomy. Science 213.4504: 161-171.
CAUTY, André
2009a Des Porteurs mayas aux éponymes aztèques’, on-line
publication: http://celia.cnrs.fr/Fr/Pub_Doc.htm#Etudes
2009b Y a-t-il des années surnuméraires mayas ?’ on-line publication
http://celia.cnrs.fr/Fr/Pub_Doc.htm#Etudes
EDMONSON, Munro S.
2000 Los calendarios de la Conquista. Arqueología mexicana VII.41:
40-45.
KIRCHHOFF, Paul
1943 Mesoamérica. Sus Límites Geográficos, Composición Étnica y
Caracteres Culturales. Acta Americana 1.1: 92-107.
MARCUS, Joyce
2000 Los calendarios. Arqueología mexicana VII.41: 12-19.
MORANTE LOPEZ, Ruben B.
2001 Las cámaras astronómicas subterráneas. Arqueología mexicana
VIII.47: 46-51.
change of synchronization tzolkin x ha’ab, or a change in the duration of a period (day, month, year,
century….) to correct the delay of the vague year over the tropical year.
51 For example: ‘experts’ (i) Mayan or (ii) Aztec vs. ‘amateurs’ (iii) Mesoamerican, mestizo or Spaniard,
of the Post Classic and Colonial Periods, etc.
CAUTY A. : Invitation to revisit the Mesoamerican Calendars…
127
POHL, Mary E., POPE, Kevin O. & NAGY, Christopher von
2002 Olmec Origins of Mesoamerican Writing. Science 6, 298.5600:
1984-1987.
ROULET, Eric
1999 Une page curieuse sur les fêtes aztèques dans un manuscrit du
XVIIIe siècle. Journal de la Société des Américanistes 85 : 367-373.
TENA, Rafael
1987 El calendario mexica y la cronografía. México: Instituto
Nacional de Antropología e Historia.
2000 El calendario mesoamericano. Arqueología mexicana VII.41: 4-11.
URCID SERRRANO, Javier
2001 Zapotec hieroglyphic writing, Studies in Pre-Columbian Art and
Archaeology 34, Dumbarton Oaks Research Library and
Collection, Washington, D.C.
Abbreviations
The names of days, months, numbers, measurement, and units of time are given in Yucatec
(colonial and latin alphabet).
Time units are written in bold underlined: tun, kin
 = integer, ranging from 0 to 13 (inclusive)
 = integer, ranging from 0 to 19 (inclusive) or from 1 to 20
X day names are in Bold (capital for the first letter): Ahau, Xochitl
x = position of the day in the cycle of 20 day names (Mayan beginning Imix)
XP = Porteur (d’année), Year Bearer
Y month names and the period of 5 days are written in Bold italic (capital for the first letter):
Pop, Uayeb, Atlcahualo
y = position in the ha'ab (Mayan beginning: Pop)
Mesoamerican figures are transcribed by their corresponding Arabic numerals. Numbers
denoting duration are transcribed like 13.0.0;0.0. where semicolon indicates the passage
between tun and uinal or like 13-baktun 0-katun 0-tun ; 0-uinal 0-kin
Pi = nodes (count) or periods (units of time measurement)
CL = Compte Long (Long Count)
CR = Calendrier Rituel (Calendar Round)
ROCm = Règle d’Orthodoxie de la Chronologie maya (4 Ahau 8 Cumku)
S = │x - │(modulo 5) is the signature of the days which belong to a same CR.
X * Y = a starry CR date whose signature is not equal to that of the origin 4 Ahau 8 Cumku.
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ANNEXES
The Mesoamerican year is still a vague solar year composed of 18 months
of 20 days, and a remainder of several days. According to the sources, the
location, or the historical period, the remainder numbers 5 or 6 days.
There are two main ways to individually determine the days of the
year. Like the Classical Mayas, sequencing them by their rank  in the Y
period, or like the Aztecs, distinguishing them by their date X in the
tonalpohualli.
DATES X OF THE 20 DAYS OF AN AZTEC MONTH (TECUILHUITL, TOVAR CODEX)
4 Cipactli
1 Ozomatli
5 Ehecatl
2 Malinalli
6 Calli
3 Acatl
7 Cuetzpalli
4 Ocelotl
8 Coatl
5 Cuauhtli
9 Miquiztli
6 Cozcacuauhtli
10 Mazatl
7 Olin
11 Tochtli
8 Tecpatl
12 Atl
9 Quiyahuitl
13 Itzcuintli
10 Xochitl
CAUTY A. : Invitation to revisit the Mesoamerican Calendars…
DATES Y OF THE 365 DAYS OF THE 19 PERIODS OF THE MAYAN HA’AB
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DATES X OF THE 20 DAYS OF THE 1st MONTH OF THE HA’AB
(Calendario de los Cakchiquel)
DATES XP OF THE 52 YEARS OF THE AZTEC XIUHTLALPILLI
(Borbonicus Codex, p. 19/21 and 20/22; XP = Tochtli, Acatl, Tecpatl, Calli)
CAUTY A. : Invitation to revisit the Mesoamerican Calendars…
DATES XP OF THE MENDOZA CODEX
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DATES X OF THE 365 DAYS OF THE 19 PERIODS OF THE HA’AB